A pond contains n fish of which 30 are carp Assume n 30 The

A pond contains n fish, of which 30 are carp. (Assume n > 30.) Then 20 fish are caught (randomly selected). Let X be the number of carp among the fish caught. Write e(n) := E[X] and v(n) := Var(X) for the expectation and variance of X (these obviously depend on n). Use indicator random variables to show e(n) = 30 20/n. That is let X_i be 1 if carp i was caught. Find a formula for v(n). (You need not simplify your formula.) Now assume the total number of fish is not a fixed number n, but is a random variable N. (Assume N > 30, always.)

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